Trigonometric Ratios and Identities: SPI Mathematics

Formulas (3)

• The six ratios of a right triangle · The six trigonometric ratios
  $$\sin A = \frac{\text{opp}}{\text{hyp}},\ \cos A = \frac{\text{adj}}{\text{hyp}},\ \tan A = \frac{\text{opp}}{\text{adj}},\ \operatorname{cosec} A = \frac{\text{hyp}}{\text{opp}},\ \sec A = \frac{\text{hyp}}{\text{adj}},\ \cot A = \frac{\text{adj}}{\text{opp}}$$
• Finding every ratio from one given ratio · Reciprocal and quotient relations
  $$\sin\theta \operatorname{cosec}\theta = 1,\quad \cos\theta \sec\theta = 1,\quad \tan\theta \cot\theta = 1,\quad \tan\theta = \frac{\sin\theta}{\cos\theta},\quad \cot\theta = \frac{\cos\theta}{\sin\theta}$$
• Dividing by cos to turn a fraction into tan · A sin and cos fraction in terms of tan
  $$\frac{a\sin\theta + b\cos\theta}{c\sin\theta + d\cos\theta} = \frac{a\tan\theta + b}{c\tan\theta + d}$$

Watch out for (6)

• Opposite and adjacent change with the angle
  → The six ratios of a right triangle
• Reading sin A as sin times A
  → The six ratios of a right triangle
• Pairing sec with sin
  → Finding every ratio from one given ratio
• Forgetting the square root
  → Finding every ratio from one given ratio
• Dividing only the top
  → Dividing by cos to turn a fraction into tan
• A plain number breaks the trick
  → Dividing by cos to turn a fraction into tan

Formulas (2)

• Reading the angle back from a value · Values that give the angle at once (0° to 90°)
  $$\sin x = \tfrac{1}{2} \Rightarrow x = 30^\circ \qquad \cos x = \tfrac{1}{2} \Rightarrow x = 60^\circ \qquad \tan x = 1 \Rightarrow x = 45^\circ \qquad \tan x = \sqrt{3} \Rightarrow x = 60^\circ \qquad \tan x = \tfrac{1}{\sqrt{3}} \Rightarrow x = 30^\circ$$
• Sides of a right triangle from one side and an angle · Legs from the hypotenuse
  $$\text{opp} = \text{hyp} \times \sin\theta, \qquad \text{adj} = \text{hyp} \times \cos\theta, \qquad \text{opp} = \text{adj} \times \tan\theta$$

Reference tables (1)

Watch out for (6)

• Doubling the angle does not double the value
  → The table of standard values
• Squaring the angle instead of the value
  → The table of standard values
• Stopping at the angle in the bracket
  → Reading the angle back from a value
• Answering the angle when a part of it is asked
  → Reading the angle back from a value
• Using sin for the side next to the angle
  → Sides of a right triangle from one side and an angle
• Taking the hypotenuse as a leg
  → Sides of a right triangle from one side and an angle

Formulas (4)

• sin²θ + cos²θ = 1 and its rearrangements · The first identity
  $$\sin^2\theta + \cos^2\theta = 1, \qquad 1 - \sin^2\theta = \cos^2\theta, \qquad 1 - \cos^2\theta = \sin^2\theta$$
• 1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ · The second and third identities
  $$1 + \tan^2\theta = \sec^2\theta, \qquad 1 + \cot^2\theta = \operatorname{cosec}^2\theta$$
• Simplifying a square root of a ratio · Roots that clear for an acute angle A
  $$\sqrt{\dfrac{1 + \cos A}{1 - \cos A}} = \operatorname{cosec} A + \cot A \qquad \sqrt{\dfrac{1 - \cos A}{1 + \cos A}} = \operatorname{cosec} A - \cot A \qquad \sqrt{1 - \sin^2 A} = \cos A \qquad \sqrt{\sec^2 A - 1} = \tan A$$
• Using an identity as a second equation · The factored identities
  $$(\sec\theta + \tan\theta)(\sec\theta - \tan\theta) = 1, \qquad (\operatorname{cosec}\theta + \cot\theta)(\operatorname{cosec}\theta - \cot\theta) = 1$$

Watch out for (8)

• The identity needs the same angle
  → sin²θ + cos²θ = 1 and its rearrangements
• The leftover goes with the bigger coefficient
  → sin²θ + cos²θ = 1 and its rearrangements
• Getting the sign the wrong way round
  → 1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ
• Pairing sec with cot
  → 1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ
• Taking the root of each term separately
  → Simplifying a square root of a ratio
• Plus and minus look alike in the options
  → Simplifying a square root of a ratio
• Trying to find the angle first
  → Using an identity as a second equation
• Adding when you need tan
  → Using an identity as a second equation

Formulas (2)

• Switching a ratio with 90° minus the angle · Complementary angle rules
  $$\sin(90^\circ - \theta) = \cos\theta,\quad \tan(90^\circ - \theta) = \cot\theta,\quad \sec(90^\circ - \theta) = \operatorname{cosec}\theta$$
• Pairing terms of a long sum or product · Number of terms in a list of angles
  $$\text{terms} = \frac{\text{last} - \text{first}}{\text{step}} + 1$$

Watch out for (4)

• Keeping the same ratio
  → Switching a ratio with 90° minus the angle
• Switching when the angles do not add to 90°
  → Switching a ratio with 90° minus the angle
• Forgetting the unpaired terms
  → Pairing terms of a long sum or product
• Miscounting the terms
  → Pairing terms of a long sum or product